Partial Differential Equations – Mathematics Notes – For W.B.C.S. Examination.

আংশিক ডিফারেনশিয়াল সমীকরণ – গণিতের নোট – WBCS পরীক্ষা।

A partial differential equation (or briefly a PDE) is a mathematical equation that involves two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables. The order of a partial differential equation is the order of the highest derivative involved. A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation. A solution is called general if it contains all particular solutions of the equation concerned.Continue Reading Partial Differential Equations – Mathematics Notes – For W.B.C.S. Examination.

General Form of First-Order Partial Differential Equation

A first-order partial differential equation with nindependent variables has the general form

F (x 1, x 2, \dots, x n, w, \partial w \partial x 1, \partial w \partial x 2, \dots, \partial w \partial x n) = 0,

where w=w(x1,x2,…,xn) is the unknown function and F(…) is a given function.

Quasilinear Equations. Characteristic System. General Solution

General form of first-order quasilinear PDE

A first-order quasilinear partial differential equation with two independent variables has the general form

f (x, y, w) \partial w \partial x + g (x, y, w) \partial w \partial y = h (x, y, w) . (1)

Such equations are encountered in various applications (continuum mechanics, gas dynamics, hydrodynamics, heat and mass transfer, wave theory, acoustics, multiphase flows, chemical engineering, etc.).

If the functions f, g, and h are independent of the unknown w, then equation (1) is called linear.

Characteristic system. General solution

The system of ordinary differential equations

d x f ( x , y , w ) = d y g ( x , y , w ) = d w h ( x , y , w ) (2)

is known as the characteristic system of equation (1). Suppose that two independent particular solutions of this system have been found in the form

u 1 (x, y, w) = C 1, u 2 (x, y, w) = C 2, (3)

where C1 and C2 are arbitrary constants; such particular solutions are known as integrals of system (2). Then the general solution to equation (1) can be written as

Φ (u 1, u 2) = 0, (4)

where Φ is an arbitrary function of two variables. With equation (4) solved for u2, one often specifies the general solution in the form u2=Ψ(u1), where Ψ(u) is an arbitrary function of one variable.

Remark. If h(x,y,w)≡0, then w=C2 can be used as the second integral in (3).

Example. Consider the linear equation

\partial w \partial x + a \partial w \partial y = b .

The associated characteristic system of ordinary differential equations

d x 1 = d y a = d w b

has two integrals

y - a x = C 1, w - b x = C 2 .

Therefore, the general solution to this PDE can be written as w−bx=Ψ(y−ax), or

w = b x + Ψ (y - a x),

where Ψ(z) is an arbitrary function.

Cauchy Problem: Two Formulations. Solving the Cauchy Problem

Generalized Cauchy problem

Generalized Cauchy problem: find a solution w=w(x,y) to equation (1) satisfying the initial conditions

x = φ 1 (ξ), y = φ 2 (ξ), w = φ 3 (ξ), (5)

where ξ is a parameter (α≤ξ≤β) and the φk(ξ) are given functions.

Geometric interpretation: find an integral surface of equation (1) passing through the line defined parametrically by equation (5).

Classical Cauchy problem

Classical Cauchy problem: find a solution w=w(x,y) of equation (1) satisfying the initial condition

w = φ (y) at x = 0, (6)

where φ(y) is a given function.

It is often convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition (6) in the parametric form

x = 0, y = ξ, w = φ (ξ) .

Existence and uniqueness theorem

If the coefficients f, g, and h of equation (1) and the functions φk in (5) are continuously differentiable with respect to each of their arguments and if the inequalities fφ′2−gφ′1≠0 and (φ′1)2+(φ′2)2≠0 hold along the curve (5), then there is a unique solution to the Cauchy problem (in a neighborhood of the curve (5)).

Procedure of solving the Cauchy problem

The procedure for solving the Cauchy problem (1), (5) involves several steps. First, two independent integrals (3) of the characteristic system (2) are determined. Then, to find the constants of integration C1 and C2, the initial data (5) must be substituted into the integrals (3) to obtain

u 1 (φ 1 (ξ), φ 2 (ξ), φ 3 (ξ)) = C 1, u 2 (φ 1 (ξ), φ 2 (ξ), φ 3 (ξ)) = C 2 . (7)

Eliminating C1 and C2 from (3) and (7) yields

u 1 (x, y, w) = u 1 (φ 1 (ξ), φ 2 (ξ), φ 3 (ξ)), u 2 (x, y, w) = u 2 (φ 1 (ξ), φ 2 (ξ), φ 3 (ξ)) . (8)

Formulas (8) are a parametric form of the solution to the Cauchy problem (1), (5). In some cases, one may succeed in eliminating the parameter ξ from relations (8), thus obtaining the solution in an explicit form.

In the cases where first integrals (3) of the characteristic system (2) cannot be found using analytical methods, one should employ numerical methods to solve the Cauchy problem (1), (5) (or (1), (6)).

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